This differs from 03._CrystalBindingAndElasticConstants.ppt

only in the section “Analysis of Elastic Strain” in which a modified version of the Kittel narrative is used.

3. Crystal Binding and Elastic Constants

• Crystals of Inert Gases

• Ionic Crystals

• Covalent Crystals

• Metals

• Hydrogen Bonds

• Atomic Radii

• Analysis of Elastic Strains

• Elastic Compliance and Stiffness Constants

• Elastic Waves in Cubic Crystals

Introduction

Cohesive energy energy required to break up crystal into neutral free atoms.

Lattice energy (ionic crystals) energy required to break up crystal into free ions.

Kcal/mol = 0.0434 eV/molecule KJ/mol = 0.0104 eV/molecule

Crystals of Inert Gases

Atoms: •high ionization energy•outermost shell filled•charge distribution spherical

Crystal: •transparent insulators•weakly bonded•low melting point•closed packed (fcc, except He3 & He4).

Van der Waals – London Interaction

Van der Waals forces ＝ induced dipole – dipole interaction between neutral atoms/molecules.

Ref: A.Haug, “Theoretical Solid State Physics”, §30, Vol I, Pergamon Press (1972).

2 2 2 2

2 1 1 2

Q Q Q QV

R

R x x R x R x

Atom i charge +Q at Ri and charge –Q at Ri + xi.( center of charge distributions )

2 22

2 1 2 1 2 1ˆ ˆ ˆ ˆ ˆ2 R x x R x R x R x R x

2 1 R R R

1/22 212R a

R a

R a 2

2

2 2

ˆ3ˆ11

2 2

a

R R R R

R aR a ˆRR R

2 2 22 1 2 1 1 22 x x x x x x

2

2 1 1 23ˆ ˆ3

QV

R R x R x x x

2

1 2 1 2 1 232

Qx x y y z z

R ˆRR z

0H H V H0 = sum of atomic hamiltonians

2

0

0 0 00 0

j

j j

VE E V

E E

0 = antisymmetrized product of ground state atomic functions

1st order term vanishes if overlap of atomic functions negligible.2nd order term is negative & R6 (van der Waals binding).

Repulsive Interaction

Pauli exclusion principle (non-electrostatic) effective repulsion

Lennard-Jones potential: 12 6

4VR R

, determined from gas phase data

/Re

Alternative repulsive term:

Equilibrium Lattice Constants

Neglecting K.E. 12 6

14

2toti j i j i j

E U Np R p R

For a fcc lattice:

12

12

112.13188

i j i jp

6

6

114.45392

i j i jp

For a hcp lattice:

12 12.13229 6 14.45489

R n.n. dist

At equilibrium:

0dE

dR

12 6

12 613 7

14 12 6

2N

R R

1/6

0 12

6

2R

1.09 for fcc lattices

Experiment (Table 4):

Error due to zero point motion

Cohesive Energy

12 6

12 6

14

2totU R NR R

26

012

1

2totU R N

26

12

48

N

4 2.15N for fcc lattices

For low T, K.E. zero point motion.

For a particle bounded within length , p

2 2

2. .

2 2

pK E

m m

1

2

quantum correction is inversely proportional to the atomic mass:~ 28, 10, 6, & 4% for Ne, Ar, Kr, Xe.

Ionic Crystals

ions: closed outermost shells ~ spherical charge distribution

Cohesive/Binding energy = 7.9+3.615.14 = 6.4 eV

Electrostatic (Madelung) Energy

Interactions involving ith ion: i i jj i

U U

2/

2

. .R

i j

i j

qn ne

RU

qotherwise

p R

tot iU NUFor N pairs of ions:2

/R qN z e

R

z number of n.n.﹦ρ ~ .1 R0

j i i jp

﹦Madelung constant

At equilibrium: 0totdU

dR

2/

2Rz q

N eR

→ 0

2/2

0R q

R ez

2

00 0

1tot

N qU

R R

2

0

N qMadelung Energy

R

Evaluation of Madelung Constant

App. B: Ewald’s method j i i jp

1 1 12 1

2 3 4

2ln 2

KCl

i fixed

Kcal/mol = 0.0434 eV/molecule Prob 3.6

Covalent Crystals

• Electron pair localized midway of bond.• Tetrahedral: diamond, zinc-blende structures.• Low filling: 0.34 vs 0.74 for closed-packed.

Pauli exclusion → exchange interaction

H2

Ar : Filled outermost shell → van der Waal interaction (3.76A)Cl2 : Unfilled outermost shell → covalent bond (2A)

s2 p2 → s p3 → tetrahedral bonds

Metals

Metallic bonding: • Non-directional, long-ranged.• Strength: vdW < metallic < ionic < covalent• Structure: closed packed (fcc, hcp, bcc)• Transition metals: extra binding of d-electrons.

Hydrogen Bonds

• Energy ~ 0.1 eV• Largely ionic ( between most electronegative atoms like O & N ).• Responsible (together with the dipoles) for characteristics of H2O.• Important in ferroelectric crystals & DNA.

Atomic Radii

Na+ = 0.97AF = 1.36ANaF = 2.33Aobs = 2.32A

Standard ionic radii ~ cubic (N=6)

Bond lengths:F2 = 1.417ANa –Na = 3.716A NaF = 2.57A

Tetrahedral:C = 0.77ASi = 1.17ASiC = 1.94AObs: 1.89A

Ref: CRC Handbook of Chemistry & Physics

Ionic Crystal Radii

E.g. BaTiO3 : a = 4.004ABa++ – O– – : D12 = 1.35 + 1.40 + 0.19 = 2.94A → a = 4.16ATi++++ – O – – : D6 = 0.68 + 1.40 = 2.08A → a = 4.16ABonding has some covalent character.

Analysis of Elastic Strains

ˆ ixLet be the Cartesian axes of the unstrained state

ix be the the axes of the stained state

Using Einstein’s summation notation, we have

ˆi j i j j xˆ ˆi i i j j x x x

1 1 11 1 11 2 k k x x 2 2 211 11 12 131 2

ˆi irr x

i ir r x

R r r ˆi i j jr x

Position of atom in unstrained lattice:

Its position in the strained lattice is defined as

Displacement due to deformation:

i j j iu r

iii ii

i

ue

x

jii j i j j i

j i

uue

x x

ˆi i ir x x ˆi iu x

Define ( Einstein notation suspended ):

i j

Dilation

1 1 2 2 3 3ˆ ˆ ˆi i i j j j k k ka b c x x x

1 1 2 2 3 3i i j j k k i j kV

1 2 3ˆ ˆ ˆV a b c abc x x x

1 2 3V x x x

211 22 33

V VO

V

211 22 331V O

1 even permutation of 123

1 for odd permutation of 123

0 otherwisei j k ijk

2123 1 23 2 1 3 3i i j j k i jkV V O

2Tr O

where

Stress Components

Xy = fx on plane normal to y-axis = σ12 .

(Static equilibrium → Torqueless) i j j i

y xX Y

Elastic Compliance & Stiffness Constants

i j i j k l k le S

1 11 xX 1 1 1

2 2 2

3 3 3

2 3 4

3 1 5

1 2 6

i j

e S

4 23 32 z yY Z

C e

S = elastic compliance tensor

Contracted indices

C = elastic stiffness tensor

Elastic Energy Density1

2U C e e

1

2C e C e U

e

C e 1

2C C e

C 1

2C C C

Let

then

1

2 i j k l i j k lU C u u

1

2C u u

Landau’s notations:

1

2ji

i jj i

uuu

x x

1

2

ii

i j

i jefor

ei j

1,2,3for

4,5,6ii

i j j i

uu

u u

u e

C

1

2C e e

Elastic Stiffness Constants for Cubic Crystals

Invariance under reflections xi → –xi C with odd numbers of like indices vanishes

Invariance under C3 , i.e.,

1111iiiiC C

x y z x x z y x

x z y x x y z x

All C i j k l = 0 except for (summation notation suspended):

1122ii k kC C 1212i k ikC C

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 1

2 2U C e e e C e e e e e e C e e e

1 111 12 12

2 212 11 12

3 312 12 11

4 444

5 544

6 644

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

eC C C

eC C C

eC C C

eC

eC

eC

11C C 12C C 44C C , 1, 2,3 4,5,6

1

11 12 12 11 12 12

12 11 12 12 11 12

12 12 11 12 12 11

44 44

44 44

44 44

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

C C C S S S

C C C S S S

C C C S S S

C S

C S

C S

where 11 12

1111 12 11 122

C CS

C C C C

12

1211 12 11 122

CS

C C C C

4444

1S

C

1

11 12 11 12S S C C 1

11 12 11 122 2S S C C

Bulk Modulus & Compressibility

2 2 2 2 2 211 1 2 3 12 1 2 2 3 3 1 44 4 5 6

1 1

2 2U C e e e C e e e e e e C e e e

Uniform dilation:

211 12

12

6U C C

1 2 3 3e e e

4 5 6 0e e e

δ = Tr eik = fractional volume change

21

2B B = Bulk modulus

1 V

V p

11 12

12

3B C C = 1/κ κ = compressibility

See table 3 for values of B & κ .

Up

2

2

UB

p

pV

V

Elastic Waves in Cubic Crystals

Newton’s 2nd law:2

2

i ki

k

u

t x

don’t confuse ui with uα

i k ik j l j lC u →2

2

jlii k j l

k

uuC

t x

221

2jl

i k j lk j k l

uuC

x x x x

2l

i k j lk j

uC

x x

2 22 2 2 2 2 23 31 1 2 2 1 1

1111 1122 1133 1212 1221 1313 13312 2 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u u uC C C C C C C

t x x x x x x x x x x x

2 22 2 2 2 23 31 2 2 1 1

1111 1122 12122 2 21 1 2 1 3 2 1 2 3 1 3

u uu u u u uC C C

x x x x x x x x x x x

22 2 2 2 2

31 1 2 1 111 12 44 442 2 2 2

1 1 2 1 3 2 3

uu u u u uC C C C

t x x x x x x x

Similarly 22 2 2 2 2

32 2 1 2 211 12 44 442 2 2 2

2 2 3 2 1 1 3

uu u u u uC C C C

t x x x x x x x

2 2 2 22 2

3 3 3 32 111 12 44 442 2 2 2

3 3 2 3 1 2 1

u u u uu uC C C C

t x x x x x x x

Dispersion Equation2 2

2i l

i k j lk j

u uC

t x x

0

i ti iu u e k r

→2

0 0i i k j l k j lu C k k u

20 0il i k j l k j lC k k u

2 0i l i k j l k jC k k dispersion equation

2 0I kC i j imn j m nC k kkC

Waves in the [100] direction

2 0I kC i j imn j m nC k kkC

1,0,0kk → 211i j i jC kkC

1111 1112 1113

22111 2112 2113

3111 3112 3113

C C C

k C C C

C C C

kC11

244

44

0 0

0 0

0 0

C

k C

C

11L

Ck

0 1,0,0u Longitudinal

44T

Ck

0 0,1,0uTransverse, degenerate 0 0,0,1u

11112

2112

3113

0 0

0 0

0 0

C

k C

C

11 16 152

61 66 65

51 56 55

C C C

k C C C

C C C

Waves in the [110] direction

2 0I kC i j imn j m nC k kkC

1,1,02

kk →

2

11 12 21 22 2i j i j i j i j i j

kC C C C kC

1111 1221 1122 12122

2121 2211 2112 2222

3113 3223

0

02

0 0

C C C Ck

C C C C

C C

kC11 44 12 442

12 44 11 44

44

0

02

0 0 2

C C C Ck

C C C C

C

11 12 44

12

2L C C C k

0 1,1,0u Lonitudinal

442T

Ck

0 0,0,1u

Transverse 1 11 12

1

2T C C k

0 1, 1,0 u

Transverse

Prob 3.10